ZERO-NONZERO PATTERNS FOR NILPOTENT MATRICES OVER FINITE FIELDS∗
KEVIN N. VANDER MEULEN† AND ADAM VAN TUYL‡
Abstract. Fix a fieldF. A zero-nonzero pattern Ais said to be potentiallynilpotent overF if there exists a matrix with entries inF with zero-nonzero pattern Athat allows nilpotence. In this paper an investigation is initiated into which zero-nonzero patterns are potentiallynilpotent overFwith a special emphasis on the case thatF=Zp is a finite field. A necessarycondition on
F is observed for a pattern to be potentiallynilpotent when the associated digraph hasm loops but no smallk-cycles, 2≤k≤m−1. As part of this investigation, methods are developed, using the tools of algebraic geometryand commutative algebra, to eliminate zero-nonzero patterns Aas being potentiallynilpotent over anyfieldF. These techniques are then used to classifyall irreducible zero-nonzero patterns of order two and three that are potentiallynilpotent overZpfor each primep.
Key words. Zero-nonzero patterns, Nilpotent, Ideal saturation, Gr¨obner basis, Finite fields.
AMS subject classifications.15A18, 13P10, 05C50, 11T06.
1. Introduction. Azero-nonzero (znz) patternAis a square matrix whose en-tries come from the set {∗,0} where ∗ denotes a nonzero entry. Fix a field F. We
then set
Q(A,F) ={A∈Mn(F)|(A)i,j= 0⇔(A)i,j=∗ for alli, j}.
The set Q(A,F),sometimes denoted Q(A) when F is known,is usually called the
qualitative classof A. An element A∈ Q(A,F) is called a matrix realization of A.
A znz-pattern A is said to be potentially nilpotent over F if there exists a matrix A ∈Q(A,F) such thatAk = 0 for some positive integer k. In this paper we study
the question of what patterns A are potentially nilpotent over a field F. Although
we will present some results for arbitrary fields,we are particularly interested in the case thatFis a finite field.
One motivation to study this question is to provide a step in understanding spectrally arbitrary patterns in the context of fields other than R. An n×n
znz-patternAis aspectrally arbitrary pattern(SAP) if given any monic polynomialp(x) of
∗Received bythe editors December 18, 2008. Accepted for publication October 8, 2009. Handling
Editor: Michael J. Tsatsomeros.
†Department of Mathematics, Redeemer UniversityCollege, Ancaster, ON L9K 1J4 Canada
(kvanderm@redeemer.ca). Research supported in part byan NSERC DiscoveryGrant.
‡Department of Mathematical Sciences, Lakehead University, Thunder Bay, ON P7B 5E1 Canada
(avantuyl@lakeheadu.ca). Research supported in part byan NSERC DiscoveryGrant.
degreenwith coefficients inF,there exists a matrixA∈Q(A,F) whose characteristic
polynomial isp(x). Note that every SAP is potentially nilpotent. There is a growing body of literature (see,for example,[2,5,7,18,19] and their references) interested in identifying SAPs whenF=R(with much focus onsign patterns: patterns whose
entries come from the set {+,−,0}). Recently,work has begun on the problem of identifying SAPs over finite fields [1] and overC[21].
We now cursorily survey the problem of identifying potentially nilpotent patterns over F=R. Determining when a sign pattern is potentially nilpotent was listed as
an open problem in [9]. Potentially nilpotent star sign patterns were introduced in [20] and fully characterized in [17]. Potentially nilpotent sign patterns of order up to 3 were characterized in [10]. Included in [10] is an investigation of sign patterns that allow nilpotence of index 2,where the index of matrixA is the smallest integer
ksuch that Ak = 0; this was later extended to index 3 in [11] (see also [3]). In [18],
it was shown that all potentially nilpotent full sign patterns (i.e. patterns with no zero entries) are also SAPs. Consequently,recent work [15] presents constructions of potentially nilpotent full sign patterns. Much work in determining when a pattern is potentially nilpotent occurs in the literature on SAPs. Identifying potentially nilpo-tent patterns over Ris in part an important subproblem in the study of SAPs due
to a technique developed in [8],usually referred to as the Nilpotent-Jacobi Method. Roughly speaking,ifAis a nilpotent realization overRof a patternA,then one can
determine if A is spectrally arbitrary by evaluating the entries of A in a Jacobian matrix constructed from A. Note that this technique requires the Implicit Func-tion Theorem,which holds overR,so one should not expect a generalization of this
approach to finite fields.
We begin in Section 2 by reviewing some basic results concerning nilpotent ma-trices over a fieldF. Many of the results that are known to hold inRcontinue to hold
over an arbitrary field.
In Section 3 we introduce some techniques to eliminate certain patterns as being potentially nilpotent over a field. We use some tools from commutative algebra and algebraic geometry to carry out this program. Starting with a znz-pattern A with nonzero entries at (i1, j1), . . . ,(it, jt),we define an ideal IA in a polynomial ring
RA=F[zi1,j1, . . . , zit,jt] over the fieldF. In Theorem 3.2 we show thatAis potentially
nilpotent over a fieldFif and only if a certain subset of the affine variety defined byIA is nonempty. With this characterization,we can use the technique of ideal saturation (see Definition 3.4) to determine if a given pattern isnotpotentially nilpotent:
Theorem 3.5. Let Fbe any field and Aa znz-pattern. Let J = (zi1,j1· · ·zit,jt)be the
Since many computer algebra programs can compute the saturation of ideals, Theorem 3.5 promises to be a useful tool for future experimentation. In the last part of Section 3 we review the basics of Gr¨obner bases,and show how Gr¨obner bases can also be used to eliminate znz-patterns as being potentially nilpotent (see Example 3.13).
As an aside,we hope that our results,along with the work of Shader [19] and Kaphle [13],will highlight the usefulness of techniques from commutative algebra and algebraic geometry in the study of SAPs. Shader uses a result about the number of algebraically independent elements over the polynomial ring R[x1, . . . , xn] to prove
a lower bound on the number of nonzero entries in a SAP. Kaphle’s MSc thesis uses Gr¨obner bases to eliminate sign patterns as being potentially nilpotent. Note that one difference between our work and the work of Kaphle is that we use the equations constructed from the characteristic polynomials when forming the Gr¨obner basis,while Kaphle uses equations constructed from the traces of the matricesAk for k= 1, . . . , n.
In Section 4 we introduce a necessary condition for a znz-patternAto be nilpotent over a field F. Precisely,we look at znz-patternsA where A is irreducible and the
digraph D(A) has no 2-cycles (see Section 2). When F = R,if A has at least two
nonzero entries on the diagonal andAis potentially nilpotent,then it is known (see [7]) that D(A) must to have a 2-cycle. We explore the fact that this is no longer true over an arbitrary field: what is important is that the polynomial x3−1 factors completely overF. In fact,we prove a more general result:
Theorem 4.4. LetAbe a znz-pattern withm≥2nonzero entries on the diagonal, and suppose that D(A)has no k-cycles with 2≤k≤m−1. IfA is potentially nilpotent overF, then the polynomial xm−1factors into mlinear forms over F.
Our paper culminates with Section 5 which uses the above techniques to classify all potentially nilpotent patterns of order at most three when F = Zp is the finite
field withpelements,wherepis a prime (see Theorems 5.1 and 5.3). One interesting by-product of this classification is the discovery thatAmay be potentially nilpotent in a field F,but asuperpattern ofA,that is,a znz-patternA′ such that (A′)i,j = 0
whenever (A)i,j= 0,might not be potentially nilpotent over the same fieldF.
2. Basic Properties. In this section,we review some of the needed properties of znz-pattern matrices and summarize some of the basic properties of potentially nilpotent matrices overF. Some of these results were known whenF=R; we consider
the more general case. We continue to use the notation from the introduction.
When referring to elements of the fieldF,we shall use 1Fto denote the
For any positive integern∈Z,we writenF to denote (1F+· · ·+ 1F) (ntimes). Then
−nF will denote the additive inverse of nF in F,and n−1F denotes the multiplicative
inverse (providednF= 0).
Given ann×nznz-patternA,we can construct a digraphD(A) = (V, E) on the vertex set V = [n] :={1, . . . , n},whose edge set consists of the arcs (i, j) whenever (A)i,j = 0. We call the edge (i, i) a loop; loops correspond to the nonzero entries on
the diagonal ofA. A simple cycle γ of length k,also called ak-cycle,is a sequence ofk distinct vertices{i1, . . . , ik} such that (i1, i2),(i2, i3), . . . ,(ik−1, ik),(ik, i1)∈E. We sometimes denote ak-cycleγby (i1, i2, . . . , ik),and denote its length by|γ|=k.
Furthermore,we say two cycles γ1 and γ2 are disjoint if they have no vertices in common.
Suppose that A ∈Q(A,F) is a realization of A. The characteristic polynomial
of A can be described in terms of the cycles of D(A). Precisely,suppose that γ = (i1, . . . , ik) is a k-cycle. We let (γ) =ai1,i2ai2,i3· · ·aik,i1 where ai,j = (A)i,j. Then
the characteristic polynomial ofAhas the form
pA(x) =xn+r1xn−1+r2xn−2+· · ·+rn−1x+rn with
ri= (−1)i (
−1)|γ1|−1(γ1)
· · ·(−1)|γp|−1(γp)
.
where the sum is over all pairwise disjoint cyclesγ1, . . . , γpsuch that|γ1|+· · ·+|γp|=i.
A znz-pattern Aof ordern≥2 is reducibleif there exists some integer 1≤r≤ n−1 and a permutation matrixP such that
PAPT =
A1 A2 0r,n−r A3
.
Otherwise,a znz-pattern A is called irreducible. Equivalently,a znz-pattern A is irreducible if and only if the associated digraphD(A) is strongly connected,that is, there is a directed path between any pair of distinct vertices. The Frobenius normal form of A is a permutation similar block upper triangular matrix whose diagonal blocks are irreducible. The diagonal blocks are called the irreducible components of A.
The final lemma of this section summarizes some of the results we will need in the later sections.
Lemma 2.1. Fix a field Fand a znz-pattern A.
(a) Suppose thatA is reducible with irreducible componentsA1, . . . ,At. ThenA
is potentially nilpotent overFif and only if each znz-patternAi is potentially
(b) If Ais potentially nilpotent over F, then so isAT, the transpose ofA.
(c) If A is a nilpotent realization of A, then the characteristic polynomial of A
ispA(x) =xn.
3. Eliminating potentially nilpotent candidates via ideal saturation.
LetF be any field. Using some tools and techniques from commutative algebra and
algebraic geometry,we will show that a znz-patternAis potentially nilpotent overF
if and only if a certain geometric set is nonempty. As an application,we develop an algebraic method to eliminate certain znz-patternsA as being potentially nilpotent overF. We also explain how to use Gr¨obner bases to show that some patterns are not
potentially nilpotent. While we will endeavor to keep this material as self-contained as possible,further background material can be found in the book of Cox,Little,and O’Shea [6].
We begin with some notation. Fix a znz-patternA,and letSA={(i, j)|(A)i,j=
0}be the locations of the nonzero elements inA. We then define the polynomial ring
RA:=F[zi,j |(i, j)∈SA] =F[zi1,j1, . . . , zit,jt]
int=|SA| variables over the fieldF. Associate toAthe matrix MAwhere
(MA)i,j:=
zi,j if (A)i,j= 0
0 if (A)i,j= 0.
Note that MA is not a realization of A since the entries of MA are variables. The characteristic polynomial ofMAthen has the form
pMA(x) =x n
−F1xn−1+F2xn−2+· · ·+ (−1)n−1Fn−1x+ (−1)nFn
where each coefficientFi =Fi(zi1,j1, . . . , zit,jt) is a polynomial in RA. We then use
thencoefficients of the characteristic polynomial to define an ideal ofRA. Precisely, let
IA:= (F1, . . . , Fn)⊆RA.
In fact, IA is a homogeneous ideal since for each Fi = 0,the polynomial Fi is a homogeneous polynomial of degree i; recall that a polynomial G is homogeneous if each term of G has the same degree. To see this fact,note that each term of Fi
corresponds to a composite cycle of length i in the directed graph D(A) (see the formula in Section 2),from which it follows that Fi is homogeneous. Hence,every znz-patternAinduces a homogeneous idealIA.
Example 3.1. We illustrate the above notation with the following znz-pattern
A=
∗ ∗ 0 ∗ 0 ∗ 0 ∗ ∗
The associated matrix is
MA=
z1,1 z1,2 0
z2,1 0 z2,3 0 z3,2 z3,3
where thezi,j’s are indeterminates in the polynomial ring
RA=F[z1,1, z1,2, z2,1, z2,3, z3,2, z3,3].
The idealIAis then generated by three homogeneous polynomials:
IA= (z1,1+z3,3, z1,2z2,1+z2,3z3,2+z1,1z3,3, z1,1z2,3z3,2+z1,2z2,1z3,3).
For eacha = (ai1,j1, . . . , ait,jt) ∈F
t,let MA(a) denote the matrix obtained by
replacing each zik,jk with aik,jk. The characteristic polynomial of MA(a) will have
the form:
pMA(a)(x) =x n
−F1(a)xn−1+F2(a)xn−2+· · ·+ (−1)n−1Fn−1(a)x+ (−1)nFn(a).
If A is potentially nilpotent over F,then there exists an a ∈ Ft with all a i,j = 0
such that MA(a) is a nilpotent matrix. In particular,the characteristic polynomial of MA(a) must be xn by Lemma 2.1 (c),which,in turn,implies thatFi(a) = 0 for
i= 1, . . . , n. Thus,one can determine if a znz-patternAis potentially nilpotent over
Fif one understands the affine variety described byIA; theaffine variety1 defined by IA is the set
V(IA) ={a∈Ft|G(a) = 0 for allG∈IA}
={a∈Ft|F1(a) =· · ·=Fn(a) = 0}.
The setV(IA) contains all the elementsa∈Ftsuch that the matrixMA(a) is
nilpo-tent. Thus,if A is potentially nilpotent over F and MA(a) is a realization of A
that is nilpotent,then a ∈ V(IA). However,the converse is not necessarily true. Indeed,if b ∈ V(IA),then while MA(b) still has a characteristic polynomial of xn,
the matrix MA(b) may not be a realization of A. As a simple example,note that 0 = (0, . . . ,0)∈V(IA),(since eachFi is homogeneous,and thus Fi(0) = 0 for all i), butMA(0) is the zero-matrix,which is not a realization ofA.
For each indeterminatezi,j∈RA,letV(zi,j) denote the associated affine variety, that is, V(zi,j) = {a ∈ Ft | ai,j = 0}. With this notation,we can determine if a
znz-patternAis potentially nilpotent overF:
1
Theorem 3.2. Fix a fieldFand a znz-patternA. ThenAis potentially nilpotent
overF if and only if
V(IA)\
t
k=1
V(zik,jk)=∅.
Proof. IfAis potentially nilpotent overF,then there exists ana∈Ft such that MA(a) is nilpotent. But that implies thata∈V(IA). Furthermore,sinceMA(a) is a realization ofA,aik,jk = 0 fork= 1, . . . , t,or in other words,a∈V(zik,jk) for each k. This proves the first direction.
For the reverse direction,ifa∈V(IA)\t
k=1V(zik,jk),thenMA(a) is a nilpotent
matrix,and furthermore,sinceaik,jk = 0 for allk,this matrix is also a realization of
A.
As a consequence of Theorem 3.2,to determine ifAis potentially nilpotent over
F,it suffices to show that the setV(IA)\t
k=1V(zik,jk) is non-empty. Unfortunately,
this can be a highly non-trivial problem. However we can use this reformulation to describe an algebraic method to determine if the setV(IA)\t
k=1V(zik,jk) is empty,
thus providing a means to determine ifAis not potentially nilpotent overF.
We begin with a simple lemma. AmonomialofRAis any polynomial of the form
m=zb1
i1,j1z
b2
i2,j2· · ·z
bt
it,jt with eachbi∈Z≥0.
Lemma 3.3. Fix a field F and a znz-pattern A. Suppose that there exists a
monomialm=zb1
i1,j1z
b2
i2,j2· · ·z
bt
it,jt ∈IA. ThenAis not potentially nilpotent over F.
Proof. For any a ∈ V(IA),we must have m(a) = ab1
i1,j1· · ·a
bt
it,jt = 0 because m∈IA. But this means thataik,jk = 0 for somek= 1, . . . , n,and thus,a∈V(zik,jk).
Now apply Theorem 3.2.
The colon operation and the saturation of ideals are two required algebraic in-gredients:
Definition 3.4. LetI andJ be ideals of a ringR. ThenI:J denotes the ideal
I:J ={g∈R|gJ ⊆I}.
Thesaturation ofI with respect toJ,denotedI:J∞,is the ideal
I:J∞={g∈R|gJi⊆I for some integeri≥0}.
Theorem 3.5. Fix a field F and a znz-pattern A. If RA =F[zi
1,j1, . . . , zit,jt],
then letmA:=tk=1zik,jk and letJ = (mA)be the ideal generated bymA. Then
(a) V(IA)\t
k=1V(zik,jk)⊆V(IA:J
∞)⊆V(I A:J);
(b) if 1 ∈ IA : J∞, then A is not potentially nilpotent over F, or any field extension of F;
(c) if1∈IA:J, thenAis not potentially nilpotent overF, or any field extension of F.
Proof. Statement (a) is a well-known result via the algebraic geometry dictionary. For completeness,we include a short proof in this context. Suppose thata∈V(IA)\
t
k=1V(zik,jk),and thus,aik,jk = 0 for k = 1, . . . , t. Suppose that G ∈ IA : J
∞. Thus,there exists an integer i such that GJi ⊆ IA. But because Ji = (miA),this implies that Gmi
A ∈ IA. Since a ∈ V(IA),we have (GmiA)(a) =G(a)miA(a) = 0. But since eachaik,jk = 0, we havem
i
A(a) =aii1,j1· · ·a
i
it,jt = 0,and henceG(a) = 0,
or equivalently,a∈V(IA:J∞). The second inclusion containment follows from the fact thatIA:J ⊆IA:J∞.
To prove (b),suppose that 1 ∈ IA : J∞. It then follows that there exists an i such thatJi ⊆IA,and hence mi
A∈IA. But then we get the desired conclusion by Lemma 3.3. In any extension ofF,we will continue to havemi
A∈IA. Statement (c) follows directly from (b) since we will have 1∈IA:J ⊆IA:J∞.
Remark 3.6. Many computer algebra systems allow one to compute the
sat-uration of an ideal,thus making Theorem 3.5 a practical tool. The computational commutative algebra programs CoCoA [4] and Macaulay 2 [12] are two free programs that can be used to compute the idealsI:J andI:J∞. On the second author’s web page2is a short introduction on how to use these programs to compute the examples found below.
Some well-known necessary facts for nilpotent matrices are simple corollaries of Theorem 3.5.
Corollary 3.7. Let A be znz-pattern. If Ahas only one nonzero entry on the
diagonal or only one transversal, thenAis not potentially nilpotent over any fieldF.
Proof. In both cases,we show that one of the generators ofIAmust be a mono-mial. If A has only one nonzero entry on the diagonal,say at position (i, i),then the trace of MA is zi,i. But since F1 = trMA = zi,i,it immediately follows that
mA ∈ IA,and hence,1 ∈ IA : (mA). Similarly,if A has only one transversal,the determinant ofMA,which equalsFn,has formzib11,j1z
b2
i2,j2· · ·z
bt
it,jt wherebk = 1 or 0.
It then follows thatmA∈IA,or equivalently,1∈IA: (mA).
2
We now provide some illustrative examples.
Example 3.8. LetA be the znz-pattern of Example 3.1. LetFbe any field of
characteristic two. Because
IA= (z1,1+z3,3, z1,2z2,1+z2,3z3,2+z1,1z3,3, z1,1z2,3z3,2+z1,2z2,1z3,3),
the monomialz21,1z3,3∈IAbecause
z1,2z2,1(z1,1+z3,3) +z1,1(z1,2z2,1+z2,3z3,2+z1,1z3,3) + (z1,1z2,3z3,2+z1,2z2,1z3,3)
= 2z1,2z2,1z1,1+ 2z1,1z2,3z3,2+z12,1z3,3+ 2z1,2z2,1z3,3=z12,1z3,3
sincex+x= 0 for anyx∈F. ThusAis not potentially nilpotent over any field of
extension ofF. Note that whenF=Z2 is the finite field with exactly two elements,
then one could use a direct calculation because there is only one choice for eachzi,j,
namely 1F. However,this method shows thatAis not potentially nilpotent over any
extension of this field.
Example 3.9. It is possible that 1∈IA:J∞,but 1∈I
A :J. As an example, consider the znz-pattern
A=
∗ 0 0 0 ∗ ∗ 0 ∗ ∗
.
We can see immediately thatAis not potentially nilpotent over any fieldFsince any
realizationA ofAwill have the nonzero eigenvalue ofa1,1. However,this cannot be
deduced fromIA:J. For example,ifF=Z2,then we use CoCoA or Macaulay 2 to findIA:J =
(z1,1+z2,2+z3,3,−z1,1z2,2+z2,3z3,2−z1,1z3,3−z2,2z3,3,−z1,1z2,3z3,2+z1,1z2,2z3,3) : (mA) = (z1,1+z2,2+z3,3, z22,2+z
2
3,3, z2,3z3,2+z2,2z3,3).
However,a computer algebra system will reveal that 1∈IA:J∞,thus showing that Ais not potentially nilpotent overF.
Example 3.10. Using the saturation of ideals also lends itself to sign patterns.
Consider the signed pattern
A=
− − − 0 0 + + + 0 0 0 0 0 − − 0 − 0 0 − − 0 0 0 0
The patternAis the patternG5studied in [14]. We then consider the matrix
MA=
−z1,1 −z1,2 −z1,3 0 0
z2,1 z2,2 z2,3 0 0 0 0 0 −z3,4 −z3,5 0 −z4,2 0 0 −z4,5 −z5,1 0 0 0 0
.
We define IA as above. LettingF=R,we find that 1∈IA: (mA)∞ using CoCoA. This sign patternA,therefore,is not potentially nilpotent overR,as first discovered
in [14]; in factG5is part of a much larger family of non-potentially nilpotent patterns.
As we will show below,the converse of Theorem 3.5 (b) does not hold. To show that Ais not potentially nilpotent,we apply the theory of Gr¨obner bases. Roughly speaking,a Gr¨obner basis ofIAis a “good” choice of generators ofIAwhich can allow one to describe the affine varietyV(IA).
We now recall the needed definitions. We fix a monomial ordering > on the monomials ofRA,that is,(1)>is a total ordering on the set of monomials,(2)>is compatible with multiplication (ifm1> m2,then for any monomialm,mm1> mm2), and (3) > is also a well-ordering. Of particular importance is the lex monomial ordering,that is,
za1
i1,j1z
a2
i2,j2· · ·z
at it,jt > z
b1
i1,j1z
b2
i2,j2· · ·z
bt it,jt
if and only if the first nonzero entry of thet-tuple (a1−b1, . . . , at−bt) is positive.
For any polynomial F =
cαmα ∈ RA where mα are monomials andcα ∈F, the leading term ofF,denoted LT>(F) is the largest monomial termcαmαinF with
respect to>.
Definition 3.11. A subset {G1, . . . , Gs} of an ideal I is a Gr¨obner basis of
I with respect to a monomial ordering > if for all F ∈ I, LT>(F) is divisible by LT>(Gi) for some i.
We then make use of the following two properties of Gr¨obner bases.
Theorem 3.12. Let R=F[zi
1,j1, . . . , zit,jt]. Let>be the lex monomial ordering
with the property that zi1,j1 >· · ·> zit,jt. LetI be an ideal of R, and suppose that
{G1, . . . , Gs} is a Gr¨obner basis ofI with respect to>. Then
(a) I= (G1, . . . , Gs), that is, the Gr¨obner basis generates I; (b) Let Il=I∩F[zi
l+1,jl+1, . . . , zit,jt]. ThenIl is thelth elimination ideal, and a
Gr¨obner basis forIl is{G1, . . . , Gs} ∩F[zil+1,jl+1, . . . , zit,jt].
To make use of the above theorem to describe the affine varietyV(I),one first finds a Gr¨obner basis{G1, . . . , Gs}forIwith respect to the lex monomial order. The-orem 3.12(b) implies that we can partition theGi’s so that the first set are
polynomi-als in the variables{zi1,j1, . . . , zit,jt},the second set are polynomials in the variables
{zi2,j2, . . . , zit,jt},and so on,i.e.,the number of variables is eliminated as you move
through the partitions. In some (but not all) cases,one or more of theGi’s may only
contain one variable. We can then find roots of these polynomials (either explicitly or numerically),and then using these solutions,find roots to the other polynomials.
We illustrate how to use Gr¨obner bases to eliminate some znz-patternsAas being potentially nilpotent over F. We will study the following pattern in more detail in
the next section.
Example 3.13. We consider the znz-pattern
A=
∗ ∗ 0 0 ∗ ∗ ∗ 0 ∗
and letF=R. In this case,the generators of the idealIAare
IA= (z1,1+z2,2+z3,3, z1,1z2,2+z1,1z3,3+z2,2z3,3, z1,2z2,3z3,1+z,1z2,2z3,3).
We can use a computer algebra program to check thatIA: (z1,1z1,2z2,2z2,3z3,1z3,3)∞=
(1). Thus Theorem 3.5 does not tell us ifAis not potentially nilpotent overR.
We use either CoCoA or Macaulay 2 to find a Gr¨obner basis forIA: {z1,1+z2,2+z3,3, z1,2z2,3z3,1+z33,3, z22,2+z2,2z3,3+z32,3}.
Notice that the last polynomial contains the fewest number of variables. If A was potentially nilpotent,then there existsa = (a1,1, a1,2, a2,2, a2,3, a3,1, a3,3)∈ R6 such
that MA(a) is nilpotent,and specifically,a is a zero of all three polynomials in the Gr¨obner basis. Notea3,3 must be a nonzero real number. But for any nonzero real numbera∈R,the last polynomial from the Gr¨obner basis implies thata2,2will then
have to satisfy
z22,2+az2,2+a2= 0⇔z2,2=a
−1±√−3 2
.
But then for every nonzero choice ofa∈R,a2,2∈R. Hence,Ais not potentially
nilpo-tent overR. Observe that this example shows that the converse of Theorem 3.5(b) is
false.
4. Digraphs without k-cycles with k small: a necessary condition. Let
nilpotent over F =R,and if D(A) has at least two loops,then D(A) must have a
2-cycle. See,for example,[7,Lemma 3.2] which considers the signed case,but the proof also holds in the non-signed case. WhenF=R,then this necessary condition
need not hold:
Example 4.1. Let A be the pattern of Example 3.13. The associated graph
D(A) has three loops but no two cycles,so [7,Lemma 3.2] implies that A is not potentially nilpotent over R. However, A is potentially nilpotent over other fields:
Bodine [1] has observed that A is SAP (hence potentially nilpotent) over certain finite fields and Yielding [21] has observed thatAis SAP overC. For example,Ais
potentially nilpotent overF=Z7 as demonstrated with the realization
4F 1F 0
0 2F 1F
−1F 0 1F
.
Our goal in this section is to explore the role of small cycles in nilpotence. More precisely,we provide a necessary condition onFfor a znz-patternAto be potentially
nilpotent overFifD(A) has loops,but nok-cycles of small size. We begin by recalling
the definition of the roots of unity and one result concerning these numbers.
Definition 4.2. Fix a fieldF. We say thatFcontains all the mthroots of unity
if all of themroots of the polynomialxm−1
F = (x−1F)(xm−1+xm−2+· · ·+x+ 1F)
belong toF,that is,xm−1
Ffactors into mlinear forms overF.
Lemma 4.3. Fix a field F and integer m ≥2. Suppose that there is a solution
(a1, . . . , am)∈Fm to the m−1 elementary symmetric polynomial equations
z1+z2+· · ·+zm= 0
z1z2+· · ·+zm−1zm= 0 .. .
z1z2· · ·zm−1+· · ·+z2z3· · ·zm= 0
with all aj= 0. ThenFcontains all the mthroots of unity.
Proof. If (a1, . . . , am) is such a solution,then (a1a−1m, . . . , ama−1m) is also a
solu-tion. Thus,we can assumeam= 1F. Hence,substituting (a1, . . . , am−1,1F) into the
above equations and rearranging gives:
a1+a2+· · ·+am−1=−1F
a1a2+· · ·+am−2am−1= 1F
.. .
We claim thata1, . . . , am−1are the remainingmthroots of unity. Indeed,
(x−a1)(x−a2)· · ·(x−am−1) =xm−1−(a1+a2+· · ·am−1)xm−2 +(a1a2+· · ·+am−2am−1)xm−3
+· · ·+ (−1)m−2(a1· · ·am−2+· · ·+a2· · ·am−1)x
+(−1)m−1
a1· · ·am−1 =xm−1+xm−2+· · ·+x+ 1F.
That is, a1, . . . , am−1 are the zeros of xm−1+xm−2+· · ·+x+ 1F. The conclusion
now follows.
Theorem 4.4. Let Abe a znz-pattern withm≥2 nonzero entries on the
diago-nal, and suppose thatD(A)has no k-cycles with 2≤k≤m−1. IfA is potentially nilpotent over F, thenF contains all themth roots of unity.
Proof. After relabeling,we may assume that the nonzero diagonal entries ofA are at (1,1), . . . ,(m, m). To simplify notation,let zi denote the variable zi,i in the polynomial ringRA. BecauseD(A) has nok-cycles with 2≤k≤m−1,this implies that the firstm−1 generators ofIAare:
F1=z1+z2+· · ·+zm F2=z1z2+· · ·+zm−1zm
.. .
Fm−1=z1z2· · ·zm−1+· · ·+z2z3· · ·zm.
Let A ∈ Q(A) be a realization that is nilpotent. If a1,1, . . . , am,m are the nonzero diagonal entries,then a = (a1,1, . . . , am,m) satisfies Fi(a) = 0 for i = 1, . . . , m−1. Because aj,j = 0 for 1≤j ≤m,Lemma 4.3 implies that the fieldF contains all the mthroots of unity.
Corollary 4.5. Let A be a znz-pattern with m ≥ 2 nonzero entries on the
diagonal, and suppose that D(A) has no k-cycles with2 ≤k ≤ m. Then A is not potentially nilpotent over any F.
Proof. We use the notation of the proof of Theorem 4.4. BecauseAhas nok-cycle with 2≤k≤m,we haveFm=z1z2· · ·zm∈IA. Now apply Lemma 3.3.
Using the above theorem,we can give a infinite family An below of potentially
nilpotent znz-patterns. In [2],this family was demonstrated to fail to be potentially nilpotent forF=R.
znz-pattern
An=
∗ ∗ 0 · · · 0 0 ∗ ∗ . .. ... ..
. . .. ... ... ... ... ..
. . .. ∗ 0
0 0 . .. ∗ ∗ ∗ 0 0 · · · 0 ∗
Then An is potentially nilpotent over F if and only ifF contains all thenth roots of
unity.
Proof. The graph ofD(A) is an n-cycle with a loop at each vertex. Thus,one direction follows immediately from Theorem 4.4 since D(A) has n loops and nok -cycles for 2≤k≤n−1. For the converse direction,suppose thatFcontains all thenth
roots of unity: supposexn−1
F= (x−ζ1)· · ·(x−ζn−1)(x−1F) withζ1, . . . , ζn−1∈F. Then the matrix
An=
ζ1 1F 0 · · · 0
0 ζ2 1F 0 0
..
. . .. ... ... . .. ... ..
. . .. . .. 0
0 0 . .. ζn−1 1F
−1F 0 0 · · · 0 1F
is a desired realization.
Corollary 4.7. Fix a prime p. If p ≡ 1 (mod n), then An is potentially
nilpotent over F=Zp.
Proof. Whenp≡1 (mod n),then by [16,Theorem 2.4],the fieldZpcontains all
thenthroots of unity. Now apply the Theorem 4.6.
Example 4.8. Theorem 4.6 gives a new way to explain why the patternA=A3
of Example 3.13 is not potentially nilpotent overR. BecauseD(A) has three loops,
but no two cycles,if A were potentially nilpotent over F,then F must contain all
the cube roots of unity. However,Rdoes not have this property. But whenF=Z7, x3−1 factors into linear forms; hence the realization in Example 4.1.
that are potentially nilpotent over F of order two or three when F = Zp,with pa
prime number. As a consequence of Lemma 2.1,it suffices to classify all znz-patterns of order two or three that are irreducible.
We begin with the 2×2 case by showing a much stronger result:
Theorem 5.1. Let F be any field. Then the znz-pattern
A=
∗ ∗ ∗ ∗
is the only irreducible 2×2 potentially nilpotent pattern overF.
Proof. The only irreducible 2×2 znz-patterns are
0 ∗ ∗ 0 ,
∗ ∗ ∗ 0 ,
0 ∗ ∗ ∗ , and
∗ ∗ ∗ ∗ .
The first three patterns cannot be potentially nilpotent overFby Corollary 3.7. The
matrix
1F 1F
−1F −1F
.
is a desired realization ofA.
The following lemma is used to shorten some of the cases in the next theorem.
Lemma 5.2. LetAbe an irreduciblen×nznz-pattern. LetD(A)be the associated
digraph.
(a) IfF =Z2 andD(A) has an odd number of loops, thenA is not potentially
nilpotent over F.
(b) If F=Z2 and D(A) has exactly two loops and two 2-cycles, then A is not
potentially nilpotent over F=Z2.
(c) The only solutions to the equation x+y+z= 0 with x, y, z∈Z3 andx,y,
z nonzero are(1F,1F,1F)and(2F,2F,2F).
Proof. (a) Suppose thatD(A) has loops at (i1, i1), . . . ,(im, im) withm= 2k+ 1. Then zi1,i1 +· · ·+zim,im ∈IA. When F=Z2,we must havezi,j = 1F for all (i, j).
But this would imply that 1F+· · ·+ 1F=mF= 0, which is false.
(b) Suppose that the diagonal entries ofA are at (i1, i1) and (i2, i2) and the two 2-cycles are (i3, j3) and (i4, j4). Then the polynomial
F2=zi1,i1zi2,i2+zi3,j3zj3,i3+zi4,j4zj4,i4 ∈IA.
IfF=Z2,then the only nonzero choice forzi,j is 1F. It then follows thatF2 cannot
(c) This statement follows from inspection.
We say that two patterns areequivalent if they have the same digraph.
Theorem 5.3. Fix a primepand an irreducible3×3znz-patternA. ThenAis
potentially nilpotent overF=Zp if and only if, up to equivalence,Aand phave one
of the following forms:
1.A=
0 ∗ 0 ∗ 0 ∗ 0 ∗ 0
∗ ∗ 0 ∗ 0 ∗ ∗ 0 ∗
0 ∗ 0 ∗ ∗ ∗ ∗ 0 ∗
or ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0
, andparbitrary.
2.A=
∗ ∗ 0 ∗ 0 ∗ 0 ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0
∗ ∗ ∗ ∗ 0 ∗ ∗ 0 ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 ∗
0 ∗ ∗ ∗ 0 ∗ ∗ ∗ 0
or ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ,
andp= 2.
3.A=
∗ ∗ 0 ∗ ∗ ∗ 0 ∗ ∗
∗ ∗ 0 ∗ ∗ ∗ ∗ 0 ∗
or
0 ∗ ∗ ∗ ∗ ∗ ∗ 0 ∗
, and p= 2 or3.
4.A=
∗ ∗ 0 0 ∗ ∗ ∗ 0 ∗
, and F=Zp contains the cube roots of unity.
Proof. We do a case-by-case analysis,by considering all 3×3 irreducible znz-patternsA. Recall that a patternAis irreducible if and only if the digraphD(A) is strongly connected. We break our proof into five cases,where each case corresponds to one of the five non-isomorphic graphs on three vertices that is strongly connected and with no loops. Each case is then broke into sub-cases,where each sub-case considers the locations of the loops.
Case 1. The non-loop edges are (1,2),(2,3),and (3,1).
In this case,we need to consider four znz-patterns:
A1,1=
0 ∗ 0 0 0 ∗ ∗ 0 0
A1,2=
∗ ∗ 0 0 0 ∗ ∗ 0 0
A1,3=
∗ ∗ 0 0 ∗ ∗ ∗ 0 0
A1,4=
∗ ∗ 0 0 ∗ ∗ ∗ 0 ∗
.
Patterns A1,1,A1,2, and A1,3 cannot be potentially nilpotent over any field F by Corollary 3.7 since one of the generators of IA1,j forj = 1,2,3 will be a monomial.
On the other hand, A1,4 is potentially nilpotent overFif and only if the polynomial
x3−1
In this case,we need to consider six znz-patterns:
A2,1=
0 ∗ 0 ∗ 0 ∗ 0 ∗ 0
A2,2=
∗ ∗ 0 ∗ 0 ∗ 0 ∗ 0
A2,3=
0 ∗ 0 ∗ ∗ ∗ 0 ∗ 0
A2,4=
∗ ∗ 0 ∗ ∗ ∗ 0 ∗ 0
A2,5=
∗ ∗ 0 ∗ 0 ∗ 0 ∗ ∗
A2,6=
∗ ∗ 0 ∗ ∗ ∗ 0 ∗ ∗
.
We can eliminate patterns A2,2,A2,3 and A2,4 as being potentially nilpotent over any Fby using Corollary 3.7. Pattern A2,1 is potentially nilpotent over any Fsince
1F,−1F∈Fand
0 1F 0
1F 0 1F
0 −1F 0
is a desired realization.
Using Theorem 3.5 and CoCoA,we can show that the pattern3
A2,5 is not po-tentially nilpotent overF=Z2 (or alternatively,see Examples 3.1 and 3.8). Ifp= 2,
thenAis potentially nilpotent overF=Zp since
1F −2−1F 0
1F 0 2−1F
0 −1F −1F
is a desired realization.
The pattern A2,6 is not potentially nilpotent over F = Z2 by Lemma 5.2 (a). Also,A2,6 is not potentially nilpotent overF=Z3. We can show this by calculating the Gr¨obner basis of IA2,6 in the ringR=Z3[z1,1, z1,2, z2,1, z2,2, z2,3, z3,2, z3,3]:
{z1,1+z2,2+z3,3, z1,2z2,1+z22,2+z2,3z3,2+z2,2z3,3+z23,3, z2,2z2,3z3,2−z2,3z3,2z3,3+z33,3}.
Sincez1,1+z2,2+z3,3= 0,by Lemma 5.2 (c),we need only consider the cases that
z1,1=z2,2=z3,3= 1F or they all equal 2F. In either case,solving for nonzero roots
of the last polynomial of the Gr¨obner basis we get z2,3z3,2−z2,3z3,2+ 1F = 0 or
2Fz2,3z3,2−2Fz2,3z3,2+ 2F = 0,neither of which has a solution inZ3. So,this pattern
is not potentially nilpotent ofZ3. On the other hand,ifp= 3, then
2F 2F 0 −4F −3F 1F
0 1F 1F
3
is a desired realization.
Case 3. The non-loop edges are (1,2),(2,1),(2,3),and (3,1).
We now need to consider eight znz-patterns:
A3,1=
0 ∗ 0 ∗ 0 ∗ ∗ 0 0
A3,2=
∗ ∗ 0 ∗ 0 ∗ ∗ 0 0
A3,3=
0 ∗ 0 ∗ ∗ ∗ ∗ 0 0
A3,4=
0 ∗ 0 ∗ 0 ∗ ∗ 0 ∗
A3,5=
∗ ∗ 0 ∗ ∗ ∗ ∗ 0 0
A3,6=
∗ ∗ 0 ∗ 0 ∗ ∗ 0 ∗
A3,7=
0 ∗ 0 ∗ ∗ ∗ ∗ 0 ∗
A3,8=
∗ ∗ 0 ∗ ∗ ∗ ∗ 0 ∗
.
Matrices A3,j for j = 1, . . . ,5 are not potentially nilpotent over any field by
Corol-lary 3.7. The matricesA3,6 andA3,7 are potentially nilpotent over any fieldF with realizations
−1F −1F 0
1F 0 1F 1F 0 1F
and
0 −1F 0
1F −1F 1F 1F 0 1F
respectively. Finally,the pattern A3,8 cannot be potentially nilpotent over Z2 by Lemma 5.2 (a). Also,this pattern is not nilpotent overZ3; again,we use a Gr¨obner
basis ofIA3,8:
{z1,1+z2,2+z3,3, z1,2z2,3+z22,2+z2,2z3,3+z32,3, z1,2z2,3z3,1+z33,3,
z22,2z2,3z3,1+z2,2z2,3z3,1z3,3+z2,3z3,1z32,3−z2,1z33,3}.
By Lemma 5.2 (c),the first polynomial implies z1,1=z2,2 =z3,3 = 1F or 2F. In the
first case,the second polynomial reduces toz1,2z2,3+ 1F+ 1F+ 1F=z1,2z2,3which is nonzero inZ3. Similarly,in the second case,the second polynomial equation becomes z1,2z2,3+ 4F+ 4F+ 4F=z1,2z2,3= 0. Ifp= 2,3,this pattern is potentially nilpotent with realization
−2F −1F 0
3F 1F 1F
1F 0 1F
.
Case 4. The non-loop edges are (1,2),(2,1),(2,3),(1,3),and (3,1).
We need to consider eight znz-patterns:
A4,1=
0 ∗ ∗ ∗ 0 ∗ ∗ 0 0
A4,2=
∗ ∗ ∗ ∗ 0 ∗ ∗ 0 0
A4,3=
0 ∗ ∗ ∗ ∗ ∗ ∗ 0 0
A4,4=
0 ∗ ∗ ∗ 0 ∗ ∗ 0 ∗
A4,5=
∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0
A4,6=
∗ ∗ ∗ ∗ 0 ∗ ∗ 0 ∗
A4,7=
0 ∗ ∗ ∗ ∗ ∗ ∗ 0 ∗
A4,8=
∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 ∗
.
We can use Corollary 3.7 to eliminate the znz-patternsA4,jforj= 1, . . . ,4. The
pat-ternsA4,j forj = 5, . . . ,8 fail to be potentially nilpotent overF=Z2 by Lemma 5.2. Indeed,the first three are eliminated by part (b),while the last is eliminated by (a).
The matricesA4,5 andA4,6are potentially nilpotent over anyF=Zpwithp= 2
with realizations
1F 1F 1F
1F −1F −1F
2F 0 0
and
1F 1F 1F
1F 0 2−1F
−2F 0 −1F
respectively.
Whenp= 3,two of the polynomials in the Gr¨obner basis ofIA4,7 arez1,2z2,1+
z3,1z1,3+z32,3 and z1,2z2,3z3,1−z1,3z3,1z3,3+z33,3. By Lemma 5.2 (c),the first poly-nomial can only equal zero ifz1,2z2,1=z3,1z1,3 =z32,3 in Z3. Since z3,3 = 0,we will always havez32,3= 1FinZ3. Hencez1,3z3,1= 1 and the second polynomial reduces to
z1,2z2,3z3,1 = 0 So,A4,7 is not potentially nilpotent over Z3. However,when p= 3, we have the realization
0 −2F 1F
1F −1F 3F·2−1F 1F 0 1F
.
The last patternA4,8 is potentially nilpotent overZp for any primep≥3 with
realization:
−2F −4F 1F
1F 1F 4−1F 1F 0 1F
.
Case 5. The non-loop edges are (1,2),(2,1),(3,2),(2,3),(1,3),and (3,1).
We now need to consider the remaining four irreducible znz-patterns:
A5,1=
0 ∗ ∗ ∗ 0 ∗ ∗ ∗ 0
A5,2=
∗ ∗ ∗ ∗ 0 ∗ ∗ ∗ 0
A5,3=
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0
A5,4=
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ .
Pattern A5,2 is not potentially nilpotent over any F by Corollary 3.7. Also,by Lemma 5.2 (a),the patternA5,4 is not potentially nilpotent overF=Z2.
nonzero solution inZ2. When p= 2,one can use the realization:
0 −1F 1F
4F 0 2F
2F 1F 0
.
The patternA5,3 is potentially nilpotent over anyF=Zp with realization:
1F 1F 1F
−1F −1F −1F
1F 1F 0
.
Finally,for anyp≥3,the patternA5,4 is potentially nilpotent overF=Zp; indeed,
one such realization is
1F 1F 1F
1F 1F 1F
−2F −2F −2F
.
Remark 5.4. We point out three interesting facts that arise from this
classifica-tion. First,all the irreducible patterns that are not potentially nilpotent over anyZp
are in fact not potentially nilpotent over any fieldF. As a consequence,to determine
which irreducible patterns are potentially nilpotent overF=R,it suffices to consider
only the irreducible patterns that appear in the statement of Theorem 5.3. Moreover, the realizations given in the proof of Theorem 5.3 show that all of these irreducible patterns are potentially nilpotent overRexceptthe pattern in Case 4. Second,ifAis
potentially nilpotent overZp,it does not necessarily follow that any superpattern of
Acontinues to be potentially nilpotent over Zp. And third,notice thatnoneof the
cases in Case 4 are potentially nilpotent overZ2. This lends itself to a natural
ques-tion: what digraphsD(A) have the property thatAfails to be potentially nilpotent over some fieldF,regardless of the placement of the loops?
Acknowledgments. Many of these results were inspired through computer exper-iments using CoCoA [4] and Macaulay 2 [12]. The second author thanks Redeemer University College for its hospitality while working on this project,while both authors thank Natalie Campbell for helpful discussions.
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